A "change of basis" is an action performed in linear algebra, whereby a change in fundamental structure yields an entirely new viewpoint. This blog began as a record of a pedagogical change of basis for me, and continues as an ongoing account of my thoughts as I design and direct courses in mathematics at the University of North Carolina, Asheville.

Wednesday, October 22, 2008

Absences are up in all of my classes (not a day goes by without a handful of students missing from Precalc, Godzilla movie marathons or no), and my students are all sick with the second major wave of colds to sweep across campus this semester. My colleagues and I wander around the department with glazed-over looks and blunder bumblingly through committee meetings.

There's more to it than a change of seasons or an onslaught of midterms. There's more to it even than the dip in the Dow and corporate profits.

I admitted to a friend the other day that I feel as though I'm suffering from an existential paralysis, and I don't think that I'm alone in that feeling. The world is holding its breath, and we're all standing here blue-faced, wondering what's going to happen next.

The moon's crawled closer to the sun each morning this week as I've made my daily trek to campus. Dawn breaks around seven or so, with blots of orange cutting through a crisp crepuscular fog. I like this time of year: the mornings are fresh, and though it's cool it's not yet cold, and I arrive at my office invigorated.

I have to admit that motivating my morning Abstract class has been a challenge for me this semester. The students are by nature quieter than those in the afternoon section, and it's often hard to rouse them. Nevertheless, they're a strong bunch, and they're quick to learn. Their committee presentations were wonderful this morning; not one of the three committees fell into the "show 'n' tell" trap of solving the problem for the rest of the class. One committee made explicit reference to the Four Cs rubric, breaking their problem down along the axes laid out by the rubric.

"Honestly, are many of you finding the Four Cs a useful tool?" I asked. "I put it out there as a tool to use, but I certainly don't want it to simply be an exercise in demagoguery. If there's anything I can do to make it better or improve upon it for future classes, let me know."

There was murmured agreement. "I use it to help critique others' work when I'm on a committee," Norbert said. "Since I'm planning on being a teacher, it really helps me to be able to grade others' work. Having some in front of me to structure my comments makes it that much easier." Others agreed, and though it sounds like no one's using the Four Cs explicitly when they're writing their own work, they're often keeping it in the backs of their minds.

The afternoon section, bigger, louder, raucous and rambunctious, offered up shorter committee reports too. They gave fewer details, and seemed a bit more timid in their responses. I wonder how much their trepidation has to do with the size of the class? I wish they'd be more bold, more descriptive. It's hard, it takes practice, but oh, how the practice is repaid!

Midway through this class the torpor returned, and I felt horribly tired. "Are you okay?" Nadia asked as class ended and people began to stow their books and take their leaves.

"I'm just really tired," I admitted.

"You don't seem tired when you're teaching," she told me. I took it as a compliment, whether it was meant that way or not. She's one of the wonderful students in our program who makes my job as pleasurable as it is.

After nearly dozing off during a frankly awful talk in the senior seminar (students: to say nothing about the poorly-organized slides and the too-detailed computations, never ever ever run over time in your talk; it's the singe worst thing you can do and while they'll forgive nearly any other sin audiences will rarely forgive you for those extra minutes), I had an a truly electrifying run home, and for the last few hours I've felt refreshed.

Just now as I was lying on the couch I gave some thought to the third chapter of my stalled series on writing pedagogy, and I promise it will come soon.

Saturday, October 11, 2008

This semester's class is a bit more balanced than last semester's. While Spring 2008's class had 10 students, all boys, this morning's class had 13 students, 6 of them girls. The better balance makes for a calmer class, as eight- and nine-year-old girls are generally a bit more staid than their masculine counterparts. They were very well-behaved today, and even though my only assistant was Zora (a high school senior who's working with me this semester in order to provide experience for her graduation project), I had no trouble at all corraling them. Moreover, the class seems like a strong one in comparison with the last couple that I've had, so I might be able to dust off the "Codemaking and Codebreaking" lesson, a fun but tough one which goes over well if the kids are up for the task and fails pretty miserably if the kids aren't quite up for the challenge.

Today it was the ever-popular "Build Your Own Fractal" lesson, in which we first play with the idea of mathematical patterns, then share in drawing some of the simplest of fractals (multiply transected triangles and squares, Sierpiński's triangle, etc.) and watch a few Mandelbrot zoom videos on YouTube before the kids go freewheeling for a half hour, designing their own self-similar patterns. We closed with twenty minutes of building pyramids, the kids (often clumsily) taping together their own three-dimensional analogues of Sierpiński's triangle.

The kids were well-behaved, attentive, shared the limited supplies (tape, glue, and scissors) responsibly, and even cleaned up after themselves at the very end without being prodded to. It was nice!

I've yet to read the final drafts of my Precalckers' poems, but I've glanced at a couple of them and noticed that some have changed significantly. Change is generally good when it comes to an iterated assignment. I'm excited to see how the students incorporated my feedback. I have a hunch there may be several very strong showings. With their permission, I'll be sure to share their work on the blog.

Okay, Maggie and I are off to a soirée at our friends' place. Au revoir!

Thursday, October 09, 2008

They're doing well. I don't think I've been challenging them enough. For instance, members of two of the six (three per section) committees who offered up reports in yesterday's classes professed that they really didn't know what to say, as the problems they'd dealt with were so straightforward. Eventually both of these teams, who worked on the same problem in different sections, delivered some variation of "It's an if-and-only-if proof; make sure you prove both directions."

One or two of the committees slipped momentarily into "show mode," essentially solving for their classmates the problem they'd been assigned. "Please remember," I gave them a gentle reminder afterward, "that it's not necessary, nor is it even best, for you to solve the problem you've been given." While one ought to offer some feedback and direction by indicating trouble spots and possible avenues to a successful proof, one does one's fellow scholars a disservice by simply solving the problem for them.

I also reminded the class that it's not up to me to weigh in the veracity of a given mathematical proposition; it's just as much up to them. Mathematics, a passel of mutually accepted axioms and rules of logic and inference, is the the product of centuries of give and take, of argument and counterargument, of disputation of every sort. Mathematics, like any other human-made system, is a social construction, and its application is open to any who master its agreed-upon conventions.

I'm liking the committees' work, and the committee system runs more smoothly in each successive class in which I make use of it. (I'm sure this in part because I'm getting better at managing and mentoring the committees, and in part because my more recent committee classes contain veterans of my earlier committee classes, so many of the students know the drill by now.) But I've definitely got to push them harder.

No, this last round didn't offer much of a hill to climb. In order to provide a greater challenge, I'm selecting some of the harder problems from the upcoming homework set for the committee problems.

Meanwhile, at other end of the Karpen Hall basement, Precalc is oozing along. We're still several sections behind where we "should be," but I'm growing more and more comfortable with that, especially after having a brief talk with my Chair this morning about my concerns. "Not having taught the class, I'm not as sure what I should be expecting them to know, and I'm not as sure of the pace I'm taking," I admitted. "It's getting better, and I'm starting to get a sense as to their abilities, but I know I'm going more slowly than I should be." He agreed with me, though, that there are certain things they need to see, and other things they can do without. As long as they're exposed to a wide variety of functions and get the chance to apply various algebraic techniques to those functions, they'll be fine.

And that they're getting.

Today's class was a tiny one, 10 of 32 people absent. (What's up, folks? Did Fall Break start early?) Class went very smoothly, though, and I noticed today how easy it now is to coax them from their seats and get them at the board doing problems. Although there are a few showboats in the class (they know who they are) who stride to the board with alacrity, even the more wallflowerish of the bunch have grown more confident in getting up in front of the class to chalk out a few figures.

All I have to say to my fellow teachers is this: if you're going to use group work, board work, inquiry-based methods, problem-based methods...any sort of classroom technique that'll take the students out of their comfort zones for even a moment...do it early, do it often. Start 'em on the first day, and do it regularly.

Indeed, this was the advice I and several others gave to Frodo, a new high school math teacher whom I met last night at a dinner meant to bring UNC Asheville's math and science faculty together with the region's high school math and science teachers. He'd related a tale of a problem-based activity he'd done in which he'd engineered the work groups by placing a strong student, a mediocre student, and a weak student in each group. The results were all right, but not what he'd hoped for. Everyone at the table assured him he'd made the right move (while the weaker students benefit from having a peer guide them through a solution, the stronger students benefit from having to provide the guidance in the first place, thereby improving their ability to communicate mathematical ideas) and encouraged him to keep at it, and to introduce such activities to the students as early in the course as possible.

I've decided that I'm going to introduce committee work to lower-level courses beginning with next semester's Calc I class.

Oh, yeah, I promised to say a bit about the Precalckers' poetic achievements! This past weekend I spent about six or seven hours reading through the 31 rough drafts I received (all but one! y'all rock!). They range from whimsical and funny to dark and brooding (one was simply titled "Dread"). Some were humorous, some wry, some philosophical. The tendency was for the poems to be more narratively personal than the Calc I students' were, and whereas the Calc I students often chose to incorporate mathematics into the structure of their poems, the Precalc students more often chose to place math squarely within the content of the poems. (This may simply be because the Calc I students have a deeper and more sophisticated understanding of mathematics in general.)

There were very few poems that I would consider weak. In fact, I would say the "low end" was significantly higher than the corresponding low for the Calc I classes last fall. On the flip side, there weren't any that approached the caliber of the Calc I students' best offerings (I'm thinking of Farrah's "Motivation for a sweet tooth," Lisette's "Mathbeth," and the anonymously penned haikus from last year's class).

This past Monday night I held an optional "poetry reading," and sadly only three students showed up. "Don't tell me we're the only ones!"Belinda moaned when she and I arrived on the scene simultaneously, finding our classroom an empty cave. "We'll give it a few minutes." A few minutes passed, and no one else had come, so Belinda and I just talked about poetry and math for a bit longer. Then Tootsie and Omar arrived, almost at once, and we got underway in earnest. Each of the three students read their poems, and we spent about ten minutes on each, offering comments and insights. Often the conversation wandered off on poorly-lit philosophical roads (Gödel's Incompleteness Theorem, the "universality" of math, ethnomathematics, cultural scientific relativism, and so forth), but I think we all learned a lot.

I feel that these poetry exercises truly help the students to engage mathematics in an entirely different way than that they're used to, and I'm hoping they're getting something meaningful out of it. Students, feel free to chime in with a comment or two! And know that I'll soon be handing out "surveys" that'll help me to understand what you got out of this exercise, and what you put into it.

I'm looking forward to hearing more from a few students in particular. For instance, not long before handing in her rough draft at the end of last week, Gwendolyn told me that she'd had a hard time finding something to write about at first, and it sounds like she spun her wheels for the first week and a half after the assignment was handed out. But then, she informed me, she'd been inspired during class last Wednesday when I'd mentioned the ways in which math could be viewed as a metaphor. The poem came quickly then.

I'd really like to get at what it was she was thinking as she put her pen to the page!

Okay, for now I must go. Tomorrow night I hope to put together Chapter 3 in my series of CWPA-inspired essays, so please stay tuned.

I plan on writing a more substantial post later from home (I'm getting off campus early today, it's been a long last few weeks!), but I wanted to check in with the latest (G,φ)-gram, this one based on the homomorphism φ from the alphabet {A,...,Z} to the group (Z,+) of integers under addition, defined by letting φ(x)=1 if x is a vowel and -1 if x is a consonant.

I'm not entirely happy with this poem, as it's a bit stilted. I admit that I wrote it largely in order to illustrate the methods of group theoretic poetics. (I'll be giving a talk on this stuff in our department's Senior Seminar next week, and I'd like to come up with a few more examples before then!) The fundamental difficulty in constructing this couplet is the prevalence of consonants in English, and I suspect writing such a balanced poem in Spanish, French, Italian, or Portuguese would be considerably easier. I believe I'll give it a try in the next few days.

Coming up in tonight's post: how are the Abstract Algebra students' committee presentations going? How are the Precalckers liking mathematical poetry, and how are they handling functions?